Theorem bnj1071 34960

Description: Technical lemma for bnj69 34993. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj1071.7	⊢ 𝐷 = (ω ∖ {∅})

Assertion

Ref	Expression
bnj1071	⊢ (𝑛 ∈ 𝐷 → E Fr 𝑛)

Proof of Theorem bnj1071

Step	Hyp	Ref	Expression
1		bnj1071.7	. . 3 ⊢ 𝐷 = (ω ∖ {∅})
2	1	bnj923 34751	. 2 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω)
3		nnord 7807	. 2 ⊢ (𝑛 ∈ ω → Ord 𝑛)
4		ordfr 6322	. 2 ⊢ (Ord 𝑛 → E Fr 𝑛)
5	2, 3, 4	3syl 18	1 ⊢ (𝑛 ∈ 𝐷 → E Fr 𝑛)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109   ∖ cdif 3900  ∅c0 4284  {csn 4577   E cep 5518   Fr wfr 5569  Ord word 6306  ωcom 7799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rab 3395  df-v 3438  df-dif 3906  df-ss 3920  df-uni 4859  df-tr 5200  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-ord 6310  df-on 6311  df-om 7800

This theorem is referenced by:  bnj1030  34970  bnj1133  34972